Finding Galaxy Clusters using Voronoi Tessellations

11 Pages·0.39 MB·English

by  M. Ramella

ASTRONOMY
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February 1, 2008
Finding Galaxy Clusters using Voronoi Tessellations
Massimo Ramella1, Walter Boschin2, Dario Fadda3, and Mario Nonino1
1 Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34100 - Trieste, Italy; [email protected], [email protected];
2 Dipartimento diAstronomia, Universit`a degli Studidi Trieste, Via Tiepolo 11, I-34100 - Trieste, Italy;[email protected];
3 Institutode Astrofisica de Canarias, ViaLactea S/N, E-38200 - La Laguna (Tenerife), Spain; [email protected]
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Received xx xxxx2000 / Accepted xxxxxx200x
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n
Abstract. We present an objective and automated pro- 1998, Boulade et al. 1998, WFI, Baade 1999, etc). The
a
J cedurefordetecting clustersofgalaxiesinimaginggalaxy possibilitytoperformwidefieldimagingoftheextragalac-
surveys1. Our Voronoi Galaxy Cluster Finder (VGCF) ticskyallowsasystematicsearchofmedium-highredshift
3
2 uses galaxypositions andmagnitudes to find clusters and galaxyclustersintwo-dimensionalphotometriccatalogsof
determine their main features: size, richness and contrast galaxies. These candidate clusters are of cosmological in-

1 abovethebackground.TheVGCFusestheVoronoitessel- terest and are primary targets for subsequent follow-up
v
lationtoevaluatethelocaldensityandtoidentifyclusters spectroscopicalobservations(see, e.g.,Holdenet al.1999,
1
assignificativedensityfluctuationsabovethebackground. Ramella et al. 2000).
1
4 Thesignificancethresholdneedstobesetbytheuser,but Several automated algorithms have already been de-
1 experimenting with different choices is very easy since it veloped for the detection of clusters within two dimen-
0 does not require a whole new run of the algorithm. The sional galaxy catalogs. The classical techniques used for
1
VGCF is non-parametric and does not smooth the data. this task are the “box count” (Lidman & Peterson 1996)
0
As a consequence, clusters are identified irrispective of and the “matched filter” algorithmproposed by Postman
/
h theirshapeandtheiridentificationisonlyslightlyaffected etal.(1996,hereafterP96)anditsrecentrefinements(see
p
bybordereffectsandbyholesinthegalaxydistributionon Kepneretal.,1999,Kawasakietal.1998,Loboetal1999).
-
o the sky. The algorithm is fast, and automatically assigns The box-counting method uses sliding windows (usu-
r members to structures. ally squares) which are moved across the point distribu-
t
s A test run of the VGCF on the PDCS field centered tion marking the positions where the count rate in the
a
: at α = 13h 26m and δ = +29o 52’ (J2000) produces 37 central part of the window exceeds the value expected
v
clusters. Of these clusters, 12 are VGCF counterparts of fromthebackgrounddeterminedintheoutermostregions
i
X the 13 PDCS clusters detected at the 3σ level and with of the window. The main drawbacks of the method are
r estimated redshifts from z = 0.2 to z = 0.6. Of the re- the introductionofabinning to determine the localback-
a maining 25 systems, 2 are PDCS clusters with confidence ground, which improves count statistics at the expense
level <3σ and redshift z ≤0.6. of spatial accuracy, and the dependence on artificial pa-
Inspectionofthe23newVGCFclustersindicatesthat rameters like bin sizes and positions or window size and
several of these clusters may have been missed by the geometry.
matched filter algorithm for one or more of the following The “matched filter” is a maximum-likelihood (ML)
reasons:a)theyareverypoor,b)theyareextremelyelon- algorithmwhichanalyzesthegalaxydistributionwiththe
gated, c) they lie too close to a rich and/or low redshift assumption of some model profiles to fit the data (e.g.
cluster. a density distribution profile and a luminosity function).
This last technique has been used to build the Palomar
Distant Cluster Survey (PDCS, see P96) catalog and the
Key words:Cosmology:large-scalestructureofUniverse
EIS cluster catalog (Olsen et al. 1999, Scodeggio et al.
– Galaxies: clusters: general – Galaxies: statistics
1999).These two catalogsare two of the largestpresently
available sets of distant clusters, with 79 and 302 candi-
1. Introduction date clusters respectively.
However, the main drawback of the matched filter
Widefieldimagingisbecomingincreasinglycommonsince
method is that it can miss clusters that are not symmet-
new large format CCD cameras are,or soon will be avail-
ric or that differ significantly from the assumed profile.
able at several telescope (see e.g. MEGACAM, Boulade
This can be a serious problem since we know that a large
Send offprint requests to: M. Ramella ([email protected]) fraction of clusters have a pronounced ellipticity (see e.g.
1 The code described in this paperis available on request Plionisetal.1991,Struble&Ftaclas1994,Wang&Ulmer
2 M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations
1997,Basilakosetal.2000).Furthermore,thematchedfil-
tertechniqueissensitivetobordereffectsbecausetheML
is computed ona circulararea.This is a problembecause
realgalaxycatalogsarefinite andusually presentholesin
regions corresponding to camera defects or bright stars.
Other interesting methods to detect overdensities in a
galaxycatalogaretheLRCFmethod(Cocco&Scaramella
1999), kernel based techniques (Silvermann 1986, Pisani
1996),and wavelettransforms (Bijaoui 1993,Fadda et al.
1997). In general, all these techniques are very sensitive
to symmetric structures and suffer from border effects.
Ebeling & Wiedenmann (1993) develop a method
based on Voronoi tessellation (VTP) in order to identify
overdensitiesinaPoissoniandistributionofphotonevents.
Their goal is to detect sources in ROSAT x-ray images.
TheVTPdoesnotsortpointsintoartificialbinsanddoes
not assume any particular source geometry for the de-
tection process.These interesting properties are also very
attractive for other applications. For example, Meurs and
Wilkinson(1999)presentanapplicationofVTPtoaslice
x (degrees)
oftheCfAredshiftsurveywiththepurposeofisolatingthe
bubbly and filamentary structure of the galaxy distribu-
tion.SeveralotherapplicationsoftheVoronoitessellation Fig.1. Voronoi tessellation of a galaxy field.
to astronomical problems have been published (see, e.g.,
van de Weygaert 2000,El-Ad & Piran 1996,1997, Ryden
cleus than to any other (see Fig. 1). For us, any point is
1995,Goldwirthetal.1995,Zaninetti1995,Doroshkevich
a galaxy of a given galaxy catalog. The algorithm used
etal.1997,Ikeuchi&Turner1991,Icke&vandeWeygaert
here for the construction of the tessellation is the “trian-
1991). Clearly, Voronoi tessellation is also interesting for
gle”Ccode byShewchuk(1996).Inorderto avoidborder
the searchof galaxy clusters (Ramella et al. 1999,Kim et
al. 2000).
In this paper we present an automatic procedure
for the identification of clusters within two-dimensional
photometric catalogs of galaxies. Our technique, an-
nounced in Ramella et al. 1999, is based on the VTP of
Ebeling & Wiedmann (1993). The procedure, hereafter
Voronoi Galaxy Cluster Finder (VGCF), is completely
non-parametric, and therefore sensitive to both symmet-
ric andelongatedand/orirregularclusters.Moreover,the
VGCF is not significantly affected by border effects, and
automatically assigns members to structures.
In Sect. 2 we describe the method and in Sect. 3 we
showanapplicationofthe VGCF tothe α=13h26m,δ =
+30o52′ field of the Palomar Distant Cluster Survey (see
P96). In particular, we illustrate the performances of the
algorithm and discuss some aspects of the its application
thatarespecific tothe constructionofa catalogofgalaxy
clusters. Finally, in Sect. 4 we summarize our results.
In what follows we assume an Hubble constant H0 =
100kms−1Mpc−1andadecelerationparameterq0 =1/2.
density
2. The method
A Voronoi tessellation on a two-dimensional distribution Fig.2. Density distribution for the points in Fig. 1 (solid
of points (called nuclei) is a unique plane partition into line) and the fitted Kiang distribution (dashed line). The
convex cells, each of them containing one, and only one, two verticallines correspondto the 95% and 99% density
nucleus and the set of points which are closer to that nu- thresholds for the selection of clusters.
M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations 3
4
3
2
1
0
16 18 20 22 24
Fig.3. The 25432 galaxies of the PDCS field centered at Fig.4. The solid line is the differential magnitude distri-
α = 201.5 degrees and δ = 29.9 degrees (J2000). Holes bution of galaxies of the PDCS cluster n. 51 (z =
PDCS
correspondtobrightstarsinthefield.Northisonthe top 0.4,N =80). The dotted line is the magnitude distribu-
R
and East on the left of the figure. tion of average galaxy counts. Counts are normalized to
an area of one square degree.
effects,werestrictourstudytoVoronoicellswhichdonot
intersect the convex hull of the set of points. In the case
In this paper, the density threshold is to separate
of galaxy catalogswith holes, we do not consider Voronoi
backgroundregions and fluctuations which are significant
cellsthatintersecttheholeborders.Todothat,wemodel
overdensities at the 80% c.l.. Since some of the selected
the hole with a combination of rectangles and we reject
overdensities can still be randomfluctuations rather than
cells which have at least one vertex inside one or more of
physical clusters, it is necessary to further suppress ran-
these rectangles.
domoverdensitiesinordertosaveonlysignificantclusters.
Since we want to find overdensities in a catalog of
We proceed as follows: we compute the probability that
galaxies, the interesting quantity for us is the density,
an overdensity corresponds to a background fluctuation
i.e. the number of galaxies for area unit, rather than the
by:
cell area associated by the Voronoi tessellation to each
galaxy.Anempiricaldistributionforrandomlypositioned n (f˜ ,0)e−bf˜minn
N(f˜ ,≤n)=N src,fluc min , (2)
points following Poissonian statistics, has been proposed min bkg b(f˜ )
min
by Kiang (1966):
where n is the number of points associated to the clus-
dp(a˜)= 44 a˜3e−4a˜da˜, (1) ter, Nbkg is the number of backgroundgalaxies,and f˜min
Γ(4)
is the minimum normalised density of the cells belong-
wherea˜≡a/<a>isthecellareainunits ofthe average ing to the cluster. Both nsrc,fluc and b are functions of
cell area < a >. As each cell contains exactly one galaxy f˜ , which can be computed from simulations (Ebeling
min
the corresponding density is the inverse of the cell area &Wiedemann,1993).Werejectoverdensitieswhoseprob-
f ≡ 1/a. The idea of Ebeling & Wiedenmann (1993) is ability to be a random fluctuation of the background is
to estimate the background by fitting the Kiang cumula- greater than 5%.
tive distribution to the empirical cumulative distribution Oncewedetect significantclusters,weregularizetheir
resulting from the real catalog in the region of low den- shape. We assume that the points inside the convex hull
sity which is not affected by the presence of structures defined by the set of points belong to the cluster itself.
(f˜≡ f/ < f >≤ 0.8). We can thus establish a density Thenwe fita circleto eachclusterandwe expandit until
threshold (Fig. 2) and define as overdensity regions those the mean density inside the circle is lower than the den-
composed by adjacentVoronoicells with a density higher sity of the original cluster. We perform this expansion of
than the chosen threshold. the original cluster because the external points of a clus-
4 M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations
4
15
3
10
2
5
1
0
0
16 18 20 22 24 0 2 4 6 8 10
Fig.5. Differential magnitude distribution of galaxies in Fig.6. Average number of false detections per field as a
acirclecenteredonasimulatedclusteratredshift0.3and function ofthe thresholdin number of coincidentfluctua-
N = 60 (solid line). The long dashed line is the mag- tions, N .
R min
nitude distribution of background galaxies and the short
dashedlineisthe magnitudedistributionofclustergalax-
3. Building a cluster catalog: the case of a PDCS
ieswithoutconsideringsurfacebrightnessselectioneffects.
field
Counts are normalized to an area of one square degree.
InthissectionwetesttheVGCFonpartofthegalaxycat-
alogusedby P96to produce the PalomarDistantCluster
terhaveusuallylargeareacells,typicalofthe lowdensity Survey (PDCS).
background. These cells can not be associated from the Inparticular,we run the VGCF onthe galaxycatalog
beginning to the cluster by the Voronoi tessellation algo- derived from the V4 band image of the PDCS field cen-
rithm. tered at α = 13h 26m and δ = +29o 52’ (J2000). The
We stress that fitting a circle to the Voronoi cluster galaxy catalog contains 25432 galaxies. For each galaxy
has absolutely no part in the detection of the cluster. It the cataloglistsrightascensionanddeclination,andtotal
onlyprovidesaconvenientwaytocatalogtheclusterwith magnitudeintheV4 band(seeP96,Sect.2).Theeffective
a center and a radius. In principle, we could fit an ellipse sky area covered by the field is 1.062 square degrees (see
rather thana circle,since our cluster finder is sensitive to Fig. 3).
elongated clusters. In practice this is not convenient be- We also have a catalog of bright stars areas to be
cause,asEbeling(1993)shows,fittedellipseeccentricities avoidedbytheVGCF.Asalreadypointedout,theVGCF
become insignificant for values lower than about 0.7. can very easily deal with “holes” in the galaxy catalog.
At this point we have a list of clusters with position,
number of members associated (we statistically subtract
3.1. Magnitude bins
the background), contrast against the background, and
projected size on the sky. We also have a file of galaxies The PDCS galaxy catalog is deep, i.e. the number den-
withposition,magnitude,areaoftheVoronoitessel,anda sity of galaxies is high. It is therefore a priori possible
label for appartenence to one of the detected clusters. As thatonlythehighestnumberdensitycontrastssurvivethe
afinal(optional)step,wecanassumeadensitymodeland “dilution”causedbyforeground/backgroundgalaxies.For
fit it on the galaxy distribution at the position where the thisreasonwedecidetoruntheVGCFinmagnitudebins.
cluster has been found. In this way we can have a better As we will show, to run the algorithm in magnitude bins
estimateofthe numberofmembersandofthe sizesofthe also helps in further suppressing spurious clusters.
cluster, in case clusters are well described by the model. The choice of the bin size is somewhat arbitrary and
cannotbeequallyoptimalforthedetectionofallclusters.
Infact,themagnituderangeoftheexcesscountsproduced
M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations 5
We choose to optimize the bin size for the detection of
PDCS-like clusters at intermediate redshift (z ∼ 0.3 −
0.6), i.e around the peak of the galaxy selection function.
In Fig. 4 we plot the magnitude distribution (normal-
ized to unit area) of galaxies within the projected area of
P51 (z ∼0.4 and richness N ∼80, see P96 for the
PDCS R
definitionofN ),superimposedontheaveragecountsper
R
squaredegreeofthewholePDCSfield.Frominspectionof
Fig.4itappearsthatareasonablechoiceforthebinsizeis
twomagnitudes.Withsuchabinmostofthecountsofthe
cluster peak fall within one bin and the cluster detection
isoptimal.Infact,awiderbinwoulddecreasethecontrast
of the cluster counts,while a narrowerbin wouldincrease
the statistical uncertainty of the detection because of the
lower number of cluster counts.
In order to further justify our bin size, we build sim-
ulated clusters at different redshifts (z =0.3, 0.5 and 0.8
respectively).Oursimulatedclustershaveaprojectedden-
sity profile with a slope r−1 outside a core radius of 0.1
Mpc h−1 and out to a cut-off radius of 1 h−1 Mpc. This
choice is motivated by the fact that PDCS clusters have,
on average, a slope r−1.4 (Lubin & Postman 1996) and
that steeper slopes produce clusters that are more easily
detected by the VGCF. We simulate the magnitudes of
cluster galaxies by sampling a Schechter luminosity func-
tion with a slope α = −1.2 and an absolute characteris-
tic magnitude M∗ = −20. We apply both evolutionary
V
andK-correctionstoclustergalaxiesaccordingto,respec-
tively, Pence (1976) and Poggianti (1996). We also take
into account selection effects in the visibility of galaxies
(Phillipsetal.1990).Wesetthe richnessofoursimulated
clustersaccordingtotheP96definitionoftherichnesspa-
rameter N (P96 clusters typically varyfrom N =10 to
R R
N = 90). As an example, in Fig. 5, we plot both the
R
countswithintheareaofasimulatedclusteratz=0.3(ra-
dius of 1 Mpc h−1 and N =60) and the average counts
R
ofthePDCSfieldexpectedwithinthesamearea.Fromthe
countdistributionofoursimulatedclusters,wefindthata
bin size of two magnitudes is indeed adequate to contain
the peak of counts corresponding to clusters within the
redshift range z ∼0.3 − 0.6.
In the case of real clusters, the position of the peak
of counts in the magnitude distribution is not known a
priori. Therefore,we“slide”atsmallstepsthe magnitude
bin over the whole magnitude range of the catalog and
runtheVGCFateachstep.We setthestepto0.1magni-
tudes, the minimum possible step considering the typical
Fig.7. Average fraction of detections of simulated clus- uncertainties of the PDCS magnitudes.
ters at z=0.3 (panel a), 0.5 (panel b) and 0.8 (panel c)
Forsimplicity,westarttheVGCFwiththebin18.0≤
as a function of N . The different curves in each panel
min
correspond to different values of the richness parameter, V4 <20.0andstopatthefaintendofthe survey,i.e.with
the bin 21.8≤V4 <23.8. We neglect 76 galaxies brighter
N .
R
thanV4 =18.0withoutconsequencesforouridentification
of clusters within the redshift range z ∼0.3 − 0.6.
by clusters varies with the cluster distance and richness.
6 M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations
Table 1. The cluster catalog
N A.R.(deg.) Decl. (deg.) R (arcsec.) Contrast Ncl Nbg Mag. bin Remarks
V1 201.623 29.977 227 7.91 25 10 4 (19.4) P55
V2 201.546 29.412 162 7.27 23 10 9 (19.9) P49
V3 201.437 29.492 133 4.47 10 5 11 (20.1) -
V4 201.595 29.559 216 9.22 69 56 31 (22.1) P50
V5 201.695 30.043 112 5.50 11 4 13 (20.3) -
V6 201.351 30.291 144 5.37 12 5 9 (19.9) -
V7 201.095 29.783 58 6.36 9 2 30 (22.0) -
V8 200.931 29.592 162 7.51 39 27 24 (21.4) P51
V9 201.381 29.640 137 9.65 32 11 26 (21.6) P52
V10 200.915 30.376 90 10.67 32 9 27 (21.7) P62
V11 201.498 30.365 155 5.81 26 20 21 (21.1) -
V12 201.247 29.720 115 6.01 19 10 22 (21.2) P53
V13 201.549 29.928 61 6.00 6 1 21 (21.1) -
V14 201.427 29.569 65 7.00 7 1 14 (20.4) -
V15 201.079 29.918 151 4.67 14 9 21 (21.1) -
V16 201.842 29.382 61 7.00 7 1 19 (20.9) P46
V17 201.910 30.094 101 6.64 23 12 28 (21.8) P58
V18 201.636 29.450 176 8.95 31 12 22 (21.2) -
V19 201.603 29.845 115 6.00 12 4 23 (21.3) -
V20 201.515 29.649 122 5.37 24 20 27 (21.7) -
V21 201.278 30.409 104 6.15 23 14 30 (22.0) P15
V22 201.733 30.348 112 7.25 29 16 27 (21.7) -
V23 200.935 29.678 108 6.50 13 4 22 (21.2) -
V24 200.957 29.451 148 6.15 23 14 22 (21.2) -
V25 201.836 29.856 68 6.05 16 7 34 (22.4) -
V26 201.941 30.184 115 6.10 22 13 29 (21.9) -
V27 201.942 29.550 72 6.35 11 3 26 (21.6) -
V28 201.776 30.005 72 5.77 10 3 28 (21.8) P56
V29 201.935 29.500 79 5.50 11 4 29 (21.9) -
V30 201.469 30.218 122 5.72 14 6 30 (22.0) -
V31 200.959 29.359 97 5.55 20 13 32 (22.2) P47
V32 202.105 30.145 115 6.01 19 10 28 (21.8) -
V33 201.866 30.140 68 5.00 10 4 28 (21.8) -
V34 200.941 30.057 86 5.81 13 5 29 (21.9) P57
V35 200.906 30.080 68 6.00 12 4 38 (22.8) -
V36 201.091 30.211 83 6.33 21 11 37 (22.7) P63
V37 200.983 30.108 111 5.63 27 23 38 (22.8) -
3.2. Clusters and their detections in bins In order to define the criterium of association for our
test application, we peform extended tests both on the
Before producing our final cluster catalog,we need to de-
PDCS field and on simulations of Poissonian fields with
fineacriteriumtoidentify clustersbyassociatingfluctua-
embedded simulated clusters. Our simulated fields have
tions detected in different magnitude bins. The criterium
the same general properties of the PDCS field. In partic-
willconsistsofamaximumprojecteddistancebetweenthe
ular, we run the VGCF on 100 catalogs each containing
centersoffluctuations tobe associatedandofaminimum
18 simulated clusters (a number similar to the number of
number of coincident fluctuations required for a positive
clustersinthe PDCSfield)embeddedwithinaPoissonian
detection,N .Infact,becauseofthestatisticalnoiseof
min galaxyfield.Clustersaresimulatedasintheprevioussub-
the foreground/background galaxy distribution, the cen-
section.Allsimulatedcatalogscontain25432galaxies,i.e.
ters of the fluctuations produced by a cluster in different
the same number of PDCS galaxies.
magnitude bins will be slightly different. As far as the
number of coincident fluctuations produced by a cluster Asaresultofourtests,weconsidercoincidenttwofluc-
is concerned, it will depend on the cluster distance, rich- tuations with centersseparatedonthe sky by a projected
ness and luminosity function. Clearly the choice of the distance d12 ≤0.3min(R1,R2), where R1 and R2 are the
criterium has to be made having in mind the goal of the radii of the two fluctuations. A tighter criterium would
detection algorithm and the characteristics of the galaxy breakthe sequenceoffluctuations correspondingto areal
catalog. cluster,aloosercriteriumwouldincorrectlyassociatefluc-
M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations 7
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
x (degrees)
Fig.8. Clusters detected by the VGCF (solid circles) and by by the matched filter algorithm (dotted circles). VGCF
identification numbers are at the center of the solid circles, PDCS identification numbers are on the side of dotted
circles. The field is centered at α= 201.5 degrees and δ = 29.9 degrees (J2000). North is on the top and East on the
left of the figure.
tuationsproducedbyadjacentclusters/fluctuationstothe creases from 1 to 5. For N = 5 there are on aver-
min
same cluster. age 1.5 spurious clusters per field. This number decreases
slowlyasN increasesfurther.Thisresultindicatesthat
We now set the minimum number of fluctuations, min
N = 5 is a a good choice to keep the number of Pois-
N , required for the detection of a cluster. In Fig. 6 min
min
son fluctuations low while still being sensitive to poor or
we plot the average number of spurious fluctuations mis-
distant clusters.
identified as clusters as a function of N . The num-
min
ber of spurious clusters drops dramatically as N in-
min
8 M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations
Fig.9. Plot of the density contrast of clusters detected by VGCF in 39 magnitude bins. Radii are scaled with the
signal-to-noise ratio of the detection. The labels on the left of the figure are the VGCF and the PDCS identification
numbers respectively.
In Fig. 7 we show the variation with N of the de- ability of detecting as a cluster a random fluctuation of
min
tection efficiency of simulated clusters at redshifts z=0.3 the galaxy distribution. For N = 5, our FPR is very
min
(panela),0.5(panelb),and0.8(panelc).Ateachredshift low. By comparison, P96 give 4.2 spurious detections per
the differentcurvescorrespondtorichnessesrangingfrom squaredegreewithpeaksignalgreaterthan3σ.Thecurves
N = 10 to N = 60. Clearly the fraction of detected in Fig. 7 correspond to a measure of the False Negative
R R
clusters decreases as N increases. Rate (FNR).
min
We note that the curve in Fig. 6 corresponds to an
evaluationoftheFalsePositiveRate(FPR),i.e.theprob-
M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations 9
6
2
4
0
2
-2
0
0 50 100 150 200 250
-2 0 2
x (arcminutes)
Fig.11. Richness distribution of P96 clusters as detected
by the VGCF (dotted line) and of new VGCF clusters
Fig.10. Plot of the galaxies of cluster V26 detected by (solid line).
the VGCF in the bin of maximum contrast before regu-
larization processes (black triangles). The grey cells are
those of galaxies added at the end of the regularization to-noise ratio, 9) the cross-identification with the PDCS
processes. Empty cells are those of background galaxies. catalog.
North is on the top and East on the left of the figure. In Fig. 8 we plot circles on the sky corresponding to
ourclusters(solidlines).We labelourclusterswitha“V”
followed by their order number in Table 1. In Fig. 9, we
3.3. The VGCF run and comparison with P96
give a graphic summary of all the fluctuations of each
We present here the results of the run of the VGCF on cluster.Theabscissaistheordernumberofthemagnitude
the V4 catalogof the PDCS field at α=13h 26m andδ = bin of the fluctuation and each cluster is represented by
+29o 52’(J2000).Asdiscussedintheprevioussection,we a row of circles parallel to the magnitude bin axis. The
runtheVGCFinbinstwomagnitudewide,“sliding”with radiiofthecirclesarescaledwiththe signal-to-noiseratio
0.1 magnitude steps within the magnitude range 18.00≤ of the detection.
V4 < 23.8. In total we run the VGCF in 39 magnitude We remind here that we fit a circle to a fluctuation
binsandidentifyasclustersatleastfivefluctuationsthat, after its detection, and that the only use of the circle is
according to our criterium, are angularly coincident (see to provide a convenient way to catalog the cluster with a
previous subsection). center and a radius.
On a Sun ULTRASparc 30 workstation, the time re- The richest and more reliable clusters exhibit, in Fig.
quired to run the whole procedure is about 40 minutes. 9, a sequence of fluctuations. The signal-to-noise ratio of
Ouroutputclusterlistconsistsof37objects.Wechar- these fluctuations regularly increases up to a maximum
acterizeeachclusterwiththepropertiesofthefluctuation andthendecreases.Severalfainterclustersshowthesame
with the highest signal-to-noise ratio as estimated by the behavior,althoughatagenerallylowerS/Nlevel.Clearly,
ratio of the number of cluster galaxies to the square root the position of the fluctuations along the magnitude axis
ofthenumberofbackgroundgalaxiesexpectedwithinthe is related to the cluster distance.
cluster area. Some clusters, for example V7 and V12, display sub-
In Table 1, for each cluster we list: 1) identification stantial gaps along the sequence. The suspicion is that
number, 2) J2000 right ascension and 3) J2000 declina- these clusters may actually consist of unrelated overden-
tion, 4) radius, 5) the cluster signal-to-noise ratio, 6) the sities projected along the same line-of-sight.
estimatednumberofclustergalaxiesand7)thenumberof We now proceed to a detailed comparison with P96
expectedbackgroundgalaxies,8)thecentralmagnitudeof clusters.TheoriginalPDCScatalogconsistsof19clusters
thebinwherewedetecttheclusterwiththehighestsignal- detected by the matched filter algorithm both in the V4
10 M. Ramella et al.: Finding Galaxy Clusters using Voronoi Tessellations
bandandinthe I4 band(see P96,Table 4).Furthermore, Thatthe generalpropertiesofnew VGCFsystemsare
the PDCS catalog lists 7 clusters detected only in the V4 not very different from P96 systems is also evident from
band and added to the catalog after visual inspection of Fig.11where we labelwith a“P”andthe PDCSidentifi-
the CCD images (see P96, Table 5). In total there are cation number the VGCF identifications of P96 clusters.
26 P96 V4-band clusters, 18 of which are detected at the AmoredetailedcomparisonbetweenourandP96clus-
3σ level.Ofthese lastclusters,13haveestimated redshift ters is not useful since, in the end, only spectroscopic ob-
within the range where our VGCF is optimal (0.2 ≤ z ≤ servations will allow a real comparison between the per-
0.6).In Fig.8we plotcirclescorrespondingto allthe P96 formances of the two algorithms. What we would like to
clusters detected by the matched filter technique (dotted stresshereisthattheVGCFisanalgorithmthatcanuse-
line). fully complement the matched filter (and probably other
We identify 12 of the 13 clusters detected by P96 at parametric cluster searches) for the definition of a com-
the 3σ level and with estimated redshifts from z =0.2 to plete and reliable 2D cluster catalog.
z =0.6.Theonlyclusterwedonotidentify,P54(z =0.5), In many problems we think that the VGCF could be
falls just under the limit of our selection criterium with preferredoverparametricsearchesand,inthesecases,the
only four coincident fluctuations. example we discuss in this section shows that the VGCF
Besidetheabove12clusters,wedetectother25galaxy retrieves most of the reliable identifications of the para-
systems.Ofthese25systems,2areP96clusterswithcon- metric searches.
fidencelevellessthan3σandredshiftz ≤0.6.Theremain-
ing 23 do not have a counterpart in the PDCS catalog.
4. Summary
Mostofthe23newclustershavepropertiescomparable
tothoseofP96clustersasdetectedbytheVGCF.Atleast
We presentanobjective andautomatedprocedureforde-
partofthenewsystemshavebeendetectedbytheVGCF
tectingclustersofgalaxiesinimaginggalaxysurveys.Our
thankstoitsspecificproperties.Forexample,becausethe
VoronoiGalaxyClusterFinder (VGCF) usesgalaxyposi-
VGCF does not smooth the data, it is able to separate
tions and magnitudes to find clustersand determine their
two large overdensities (V2 and V18) that are unlikely to
main features: size, richness and contrastabove the back-
be one system(P49).InfactV2 (whichwe identify asthe
ground. The VGCF uses the Voronoi tessellation to eval-
counterpart of P49) and V18 are well separated both on
uate the local density and to identify clusters as signi-
the sky and in magnitude.
ficative density fluctuations above the background. The
The property of the VGCF of detecting overdensities significance threshold needs to be set by the user, but
irrespective of their shape is probably the key for the de- experimenting with different choices is very easy since
tectionofV26,averyelongatedsystem.InFig.10weshow it only requires a new selection from the output of the
the galaxies of V26 as detected by the VGCF before the Voronoitessellation and not a whole new run of the algo-
regularization processes (black triangles). We also show rithm.TheVGCFisnon-parametricanddoesnotrequire
as grey tessels those of galaxies added at the end of the a smoothing of the data. As a consequence, clusters are
regularizationprocessesandaswhitetesselsthoseofback- identified irrispective of their shape and their identifica-
ground galaxies. The plot refers to the tessellation of the tionisonlyslightlyaffectedbybordereffectsandbyholes
PDCSfieldinthemagnitudebinofmaximumcontrastfor in the galaxy distribution on the sky.
V26.
Thealgorithmisfast,andautomaticallyassignsmem-
As far as the richness, NR, of VGCF clusters is con- bers to structures. For example, a run on about 25000
cerned, we compute it almost in the same way as P96, galaxies only requires 7 minutes on a Sun ULTRASparc
the only difference being that we use a fixed angular ra- 30 workstation.
dius of0.1degwithin whichtocountclustergalaxies.For We perform test runs of the VGCF both onsimulated
systems within the redshift range 0.3-0.6, our angular ra- galaxy fields and on the 25432 galaxies identified in the
diusroughlycorrespondstoalinearradiusof1 h−1 Mpc. V4 band image of the PDCS field centered at α = 13h
As P96 shows, NR is related to the Abell richness of the 26m and δ = +29o 52’ (J2000, see P96). Given the depth
system. of the PDCS survey, we run the VGCF in 39 overlapping
In Fig. 11 we plot the distribution of NR of our new 2 magnitude wide bins, from V4 = 18.00 to V4 = 23.75.
systems (solid histogram)and of P96 clusters as detected For the detection of a cluster we require at least five an-
by the VGCF (dotted histogram). In general the richness gularly coincident fluctuations. Based on 100 simulations
of the new systems is similar to that of P96 clusters (as of the PDCS field, we find that there are, on average,1.5
detected by the VGCF), but the VGCF detects a larger spurious detections in a simulated PDCS field with Pois-
fractionofpoorsystemsthanthematchedfilteralgorithm. son background. Our choice of bin size and number of
In fact, we have 5 systems with N < 20 while only 1 bins, together with the minimum number of fluctuations
R
P96clusterfallswithinthisrange.Weprobablyfindthese required for a cluster detection, optimizes the VGCF for
poorer systems because we do not smooth the data. the redshift range 0.3∼<z ∼<0.6.